(1.2.1) E(Y1Y2 Fz) = E(YIIF )E(Y21Fz).
This condition is equivalent [4, II.45] to
(1.2.2) E(Y1IF ) = E(Y IFz)
for every F -measurable, Integrable r.v. Y1. The condition (F4) can
z
be seen to be equivalent to the condition of commutation as follows:
(F4) => commutation: Let X be F-measurable, integrable. Then
E(XIF1) Is F1-measurable, integrable. By (F4), E(E(XIF1)IF2)
E(E(XIFz)IFz) = E(XIFz). Similarly, E(XIF2) is F2-measurable,
integrable; hence by (F4) E(E(X F2) F) = E(E(X F2)I F) E(XIFz).
Thus E(E(XF1 )IF2) = E(XIFz) = E(E(X F2) Fz), i.e., the operators
E(.IF2) and E(.IF1) commute.
Commutation => (F4): We shall assume commutation and prove that
(1.2.2) holds.
Let Y be F -measurable, integrable. We have, from the
1 z
commutation condition,
E(Y jF) = E(YjFz F2) 2 E(Y1 2 )
which Is just (1.2.2).
Although the difference between the two constructions is slight
(the main difference being on the "border at infinity," where we do
not necessarily have right continuity of the filtrations (F ) and
(F ) in the Cairoli-Walsh model), the condition most often imposed on
t
the filtration in the literature is stated as the (F4) condition,
although the notation used is more often that of Meyer.